Focal Underdetermined System Solver Research Articles

2D Beam Domain Statistical CSI Estimation for Massive MIMO Uplink

In this paper, we investigate the beam domain statistical channel state information (CSI) estimation for the two-dimensional (2D) beam-based statistical channel model (BSCM) in massive multi-input multi-output (MIMO) systems. The problem is to estimate the beam domain channel power matrices (BDCPMs) based on multiple received pilot signals. A received signal model showing the relation between the statistical properties of the received pilot signals and the BDCPMs is derived. On the basis of the received signal model, we formulate an optimization problem with the Kullback-Leibler (KL) divergence. By solving the optimization problem, a novel method to estimate the statistical CSI without the estimates of instantaneous CSI is proposed. We further reduce the complexity of the proposed method by utilizing the circulant structures of particular matrices in the algorithm. We also showed the generality of the proposed method by introducing another application, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i.e</i> ., estimation of the angle domain channel power matrix. Simulation results show that the proposed method has good convergence and can obtain sparse BDCPMs. Compared with the regularized multiple measurement vector focal underdetermined system solver (RM-FOCUSS) algorithm, the proposed algorithm obtains overall more accurate statistical CSI with much lower complexity and brings significant performance gains when used in channel estimation.

Real-Valued Deep Unfolded Networks for Off-Grid DOA Estimation via Nested Array

Recently, deep unfolded networks have been widely utilized in direction of arrival (DOA) estimation due to the reduced computational complexity and improved estimation accuracy. However, few consider the nested array for off-grid DOA estimation, where the estimated DOAs are not on the prespecified grids. In this paper, we propose a deep unfolded FOCal Underdetermined System Solver (FOCUSS) network and a deep unfolded Alternating Direction Method of Multiplies (ADMM) to address the problem, which respectively aim to improve estimation accuracy and further reduce computational complexity. We first apply first-order Taylor expansion and vectorize the covariance matrix into a real-valued single snapshot for network input. We then train the proposed networks to obtain on-grid DOA spatial spectrum and off-grid values, where the off-grid DOA estimation is calculated by the peaks of on-grid DOA spatial spectrum and corresponding off-grid values. We demonstrate that the proposed networks with interpretable parameters can accelerate the convergence rate and achieve better generalization. Simulations verify the performance of proposed networks in comparison with the existing methods.

Direction finding by sparse‐based approach for dual‐polarised conformal array in the presence of gain and phase uncertainties

AbstractThis article investigates the problem of robust direction‐of‐arrival (DOA) estimation with unknown sensor gain and phase uncertainties. A Dual‐Polarised Regularised Multiple FOCal Underdetermined System Solver is proposed to estimate the 2‐D DOA for the dual‐polarised conformal array. Each polarisation signal is decomposed into two orthogonal components, which are regarded as a pair of coherent signals, and the polarisation parameters are merged into the signal waveform. A sparse‐based method based on the coherent signals block is presented, implementing weight sharing and grid pruning within the block to alleviate the negative influence of the errors. Meanwhile, gain and phase errors are modelled as the equivalent noise, which can be suppressed with the adaptive adjustment for the regularisation parameter. As the insensitivity of the sparse‐based algorithm to coherent signals, the proposed algorithm can realise the blind polarisation 2‐D DOA estimation. The Cramer–Rao bound is derived, and simulations exhibit the effectiveness and robustness of the proposed method with the unknown sensor gain and phase uncertainties.

Fully cooperative and distributed focal underdetermined system solver compressive sensing recovery algorithm for wireless sensor networks

SummaryThe designed algorithms for wireless sensor networks (WSNs) should not impose a high computational load on sensor nodes because they are battery‐powered devices. To distribute the computations between sensors, distributed algorithms are required. On the other hand, since the sensors observe related phenomena, the acquired signals, besides the intra‐signal correlation, also have some inter‐signal correlation. When the designed algorithms are cooperative, these joint structures are used. In this study, we propose a distributed and cooperative algorithm based on the focal underdetermined system solver (FOCUSS) for compressive sensing signal reconstruction in WSNs. Unlike the other FOCUSS‐based sparse recovery algorithms, the proposed algorithm is quite distributed and implemented by one‐hop communication between the neighboring node without collaborating any fusion center (FC). In the suggested system model, the reconstructed signals from the underdetermined systems of equations are different for each sensor node. The obtained results verified that the proposed approach performs better than the existing works regarding signal reconstruction and convergence rate.

Method of Range Ambiguity Suppression Combining Sparse Reconstruction and Matched Filter

Synthetic Aperture Radar (SAR) images are often affected by range ambiguity due to antenna sidelobe characteristics and pulse operating mechanism. The work of range ambiguity suppression focuses on both SAR system design and signal processing. On the one hand, ideas have been proposed to modify the transmitting and receiving waveforms to block the ambiguous energy, such as up and down chirp modulation, multiple elevation beams, and azimuth phase coding techniques. On the other hand, signal processing methods in echo and image domains have been used to reduce the range ambiguity energy, such as up and down chirp processing and sparse regularization methods. This paper proposes a range ambiguity suppression method that combines sparse reconstruction and matched filter. There are four main steps: performing the sparse reconstruction to obtain the ambiguity area, estimating the ambiguity signal using the ambiguity area image and reconstruction matrix, separating ambiguity signal from the whole signal to obtain the signal of primary imaging area, using matched filter to the primary signal to obtain the main lobe area image. In this method, sparse reconstruction improves the accuracy of ambiguity signal estimation, and matched filter ensures the efficiency of imaging processing procedure. Simulation results show that the proposed method can effectively suppress range ambiguity. When the ambiguity energy is relatively strong, the Focal Underdetermined System Solver (FOCUSS) algorithm can get better results than the classical Orthogonal Matching Pursuit (OMP) algorithm. Moreover, the weak targets and the details of the main lobe image are well preserved.

Multi-coset angular sampling-based compressed sensing of blade tip-timing vibration signals under variable speeds

Blade Tip-Timing (BTT) has been regarded as a promising way of on-line blade vibration monitoring. But blind multi-band BTT vibration reconstruction is a big challenge under variable speeds and under-sampling. In order to deal with it, a novel Compressed Sensing (CS) method is proposed based on Multi-Coset Angular Sampling (MCAS) in this paper. First, multi-coset sampling scheme of BTT vibration signals is presented. Then the CS model of BTT vibration signals is derived in order domain. A sufficient condition of the number of BTT sensors is derived for perfect reconstruction and optimal placement of BTT sensors is determined by minimizing the condition number. In the end, numerical simulations are done to validate the proposed method and the performances of four reconstruction algorithms are compared, i.e., Orthogonal Matching Pursuit (OMP), Multiple Signal Classification (MUSIC), Basis Pursuit Denoising (BPDN) and Modified Focal Underdetermined System Solver (MFOCUSS). Influences of the sensor placement, the number of BTT sensors and measurement noises on the reconstruction performances are also testified. The results demonstrate that the proposed method is feasible and the overall performance of the BPDN algorithm is the best among the four algorithms. Also the reconstruction performance decreases with the accelerations of rotating speed.

AFODSS: Heart Rate Estimation Method From Photoplethysmographic Signals With Motion Artifacts Using Fourier-Sparse Dual Optimization

Precise heart rate (HR) estimation of individuals during physical exercise from PPG sensors using direct methods often produces erroneous output due to the occurrence of exceedingly high motion artifacts (MAs). The effect of MAs can be neutralized using non-direct algorithms such as optimization techniques using some a priori about MA. In this paper, we propose a dual-optimization technique using adaptive Fourier decomposition in conjunction with sparse spectral reconstruction (AFODSS) using a multiple measurement vectors (MMVs) model and a regularized bi-conjugate gradient focal underdetermined system solver (BCG-FOCUSS) algorithm. Using AFODSS, the average absolute error (AAE) of HR obtained on the dataset recorded from 23 subjects during their intense exercise is 1.35 beats/min. The BCG-FOCUSS algorithm used in this paper helps to speed up the convergence rate and reduce time complexity when compared with other FOCUSS algorithms.

Improved Reconstruction of MR Scanned Images by Using a Dictionary Learning Scheme.

The application of compressed sensing (CS) to biomedical imaging is sensational since it permits a rationally accurate reconstruction of images by exploiting the image sparsity. The quality of CS reconstruction methods largely depends on the use of various sparsifying transforms, such as wavelets, curvelets or total variation (TV), to recover MR images. As per recently developed mathematical concepts of CS, the biomedical images with sparse representation can be recovered from randomly undersampled data, provided that an appropriate nonlinear recovery method is used. Due to high under-sampling, the reconstructed images have noise like artifacts because of aliasing. Reconstruction of images from CS involves two steps, one for dictionary learning and the other for sparse coding. In this novel framework, we choose Simultaneous code word optimization (SimCO) patch-based dictionary learning that updates the atoms simultaneously, whereas Focal underdetermined system solver (FOCUSS) is used for sparse representation because of a soft constraint on sparsity of an image. Combining SimCO and FOCUSS, we propose a new scheme called SiFo. Our proposed alternating reconstruction scheme learns the dictionary, uses it to eliminate aliasing and noise in one stage, and afterwards restores and fills in the k-space data in the second stage. Experiments were performed using different sampling schemes with noisy and noiseless cases of both phantom and real brain images. Based on various performance parameters, it has been shown that our designed technique outperforms the conventional techniques, like K-SVD with OMP, used in dictionary learning based MRI (DLMRI) reconstruction.

A Novel 2D Off-Grid DOA Estimation Method Based on Compressive Sensing and Least Square Optimization

For two-dimensional (2D) direction of arrival (DOA) estimation in a uniform rectangular array (URA), the conventional method converts the 2D problem into a one-dimensional (1D) problem; however, the computational complexity is high, and the accuracy is limited by the grid interval. To address this issue, based on sparse representation theory and a separable observation model (SPM), this paper presents a novel off-grid-framework-based 2D DOA estimation approach by designing a modified 2D off-grid model and a solution for the multisnapshot case in the SPM. The proposed algorithm can be divided into two stages. In the first stage, we use a matching pursuit and focal underdetermined system solver (MFOCUSS) algorithm to quickly identify the candidate or potential areas where the true sources may exist. In the second stage, the candidate areas obtained in the first stage are regarded as the initialization. Then, for a specific source, we regard other sources as interference. By using an alternating descent method, we can obtain accurate DOAs. Moreover, based on the equivalence of time delay and spatial spacing, a 2D off-grid method for multisnapshot cases is proposed in this paper. Numerical simulations demonstrate the effectiveness and efficiency of the proposed algorithm.

A Novel Underdetermined Source Recovery Algorithm Based on k-Sparse Component Analysis

Sparse component analysis (SCA) is a popular method for addressing underdetermined blind source separation in array signal processing applications. We are motivated by problems that arise in the applications where the sources are densely sparse (i.e. the number of active sources is high and very close to the number of sensors). The separation performance of current underdetermined source recovery (USR) solutions, including the relaxation and greedy families, reduces with decreasing the mixing system dimension and increasing the sparsity level (k). In this paper, we present a k-SCA-based algorithm that is suitable for USR in low-dimensional mixing systems. Assuming the sources is at most $$(m-1$$ ) sparse where m is the number of mixtures; the proposed method is capable of recovering the sources from the mixtures given the mixing matrix using a subspace detection framework. Simulation results show that the proposed algorithm achieves better separation performance in k-SCA conditions compared to state-of-the-art USR algorithms such as basis pursuit, minimizing norm-L1, smoothed L0, focal underdetermined system solver and orthogonal matching pursuit.

Fast BCS-FOCUSS and DBCS-FOCUSS with augmented Lagrangian and minimum residual methods

Block compressive sensing FOCal Underdetermined System Solver (BCS-FOCUSS) and distributed BCS-FOCUSS (DBCS-FOCUSS) are iterative algorithms for individual and joint recovery of correlated images. The performance of both these algorithms was noticed to be best within BCS framework. However, both these algorithms suffer from high computational complexity and recovery time. This is caused by the need for an explicit computation of matrix inverse in each iteration and a slow convergence from a poor starting point. In this paper, we propose a methodology to obtain fast and good initial solution using the augmented Lagrangian method to improve the convergence rate of both algorithms. We also propose to incorporate the minimum residual method to avoid matrix inversion to reduce the computational cost. Simulation studies with the proposed modified BCS-FOCUSS and DBCS-FOCUSS demonstrate a significant reduction in the computational cost and recovery time while improving reconstruction quality for both individual and joint reconstruction algorithms.

Robust nonhomogeneous training samples detection method for space–time adaptive processing radar using sparse-recovery with knowledge-aided

The performance of space–time adaptive processing (STAP) may degrade significantly when some of the training samples are contaminated by the signal-like components (outliers) in nonhomogeneous clutter environments. To remove the training samples contaminated by outliers in nonhomogeneous clutter environments, a robust nonhomogeneous training samples detection method using the sparse-recovery (SR) with knowledge-aided (KA) is proposed. First, the reduced-dimension (RD) overcomplete spatial–temporal steering dictionary is designed with the prior knowledge of system parameters and the possible target region. Then, the clutter covariance matrix (CCM) of cell under test is efficiently estimated using a modified focal underdetermined system solver (FOCUSS) algorithm, where a RD overcomplete spatial–temporal steering dictionary is applied. Third, the proposed statistics are formed by combining the estimated CCM with the generalized inner products (GIP) method, and the contaminated training samples can be detected and removed. Finally, several simulation results validate the effectiveness of the proposed KA-SR-GIP method.

Solution Strategies for Linear Inverse Problems in Spatial Audio Signal Processing

The aim of this study was to compare algorithms for solving inverse problems generally encountered in spatial audio signal processing. Tikhonov regularization is typically utilized to solve overdetermined linear systems in which the regularization parameter is selected by the golden section search (GSS) algorithm. For underdetermined problems with sparse solutions, several iterative compressive sampling (CS) methods are suggested as alternatives to traditional convex optimization (CVX) methods that are computationally expensive. The focal underdetermined system solver (FOCUSS), the steepest descent (SD) method, Newton’s (NT) method, and the conjugate gradient (CG) method were developed to solve CS problems more efficiently in this study. These algorithms were compared in terms of problems, including source localization and separation, noise source identification, and analysis and synthesis of sound fields, by using a uniform linear array (ULA), a uniform circular array (UCA), and a random array. The derived results are discussed herein and guidelines for the application of these algorithms are summarized.

Improved focal underdetermined system solver method for radar coincidence imaging with model mismatch

Radar coincidence imaging (RCI) is a staring imaging technique that originated from optical coincidence imaging. In RCI, the reference matrix needs to be computed precisely to reconstruct the image. However, it is difficult to exactly calculate the reference matrix as model mismatch existing in most applications. The signal model of RCI with model mismatch is derived. Based on a Bayesian framework and regularization method, an algorithm called regularization-focal underdetermined system solver (R-FOCUSS) is proposed to solve the RCI problem with model mismatch. In the proposed method, the scattering coefficients and the perturbation matrix can be calculated during the iterations, so the image can be reconstructed. A norm-ratio method is also proposed to determine the regularization parameters in the objective function, which makes the algorithm suitable for the situation, where the distributions of noise, model error, and target’s sparsity are unknown. The constrained Cramer–Rao bound for scatterer estimation is derived. Compared with some existing sparse reconstruction methods, R-FOCUSS is more robust, with a lower computation complexity. Results of numerical experiments demonstrate that the algorithm can achieve outstanding imaging performance and yields superior performance both in suppressing noise and in adapting to model mismatch.

A multilayer FOCUSS approach for sparse representation

Focal Underdetermined System Solver (FOCUSS) is a powerful method for sparse representation, in which the Lp-norm like cost function is very often used. However, this cost function is not only nondifferentiable but also can be very ill-conditioned in some situations. The local minima problem of FOCUSS is discussed in this paper. Moreover, to solve this problem, we first extend the Lp-norm like cost function to its corresponding Lp-approximation. After this, we analyze the nonconvexity of the new cost function, which results in that FOCUSS algorithm gets stuck in the local minima in many situations, especially when the hidden sources are not very sparse. To reduce the number of the local minima, a multilayer FOCUSS is developed in this paper. Comparing with the conventional FOCUSS, the experiments inclusing MRI reconstruction demonstrate that multilayer FOCUSS can significantly improve the performance. Even for some very challenging cases, where the conventional FOCUSS fails, multilayer FOCUSS still works well.

Fast FOCUSS method based on bi-conjugate gradient and its application to space-time clutter spectrum estimation

The focal underdetermined system solver (FOCUSS) is a powerful tool for sparse representation in complex underdetermined systems. This paper presents the fast FOCUSS method based on the bi-conjugate gradient (BICG), termed BICG-FOCUSS, to speed up the convergence rate of the original FOCUSS. BICGFOCUSS was specifically designed to reduce the computational complexity of FOCUSS by solving a complex linear equation using the BICG method according to the rank of the weight matrix in FOCUSS. Experimental results show that BICG-FOCUSS is more efficient in terms of computational time than FOCUSS without losing accuracy. Since FOCUSS is an efficient tool for estimating the space-time clutter spectrum in sparse recoverybased space-time adaptive processing (SR-STAP), we propose BICG-FOCUSS to achieve a fast estimation of the space-time clutter spectrum in mono-static array radar and in the mountaintop system. The high performance of the proposed BICG-FOCUSS in the application is demonstrated with both simulated and real data.

Synthesis of Pattern Reconfigurable Sparse Arrays With Multiple Measurement Vectors FOCUSS Method

A new technique based on the multiple measurement vectors FOCal Underdetermined System Solver (M-FOCUSS) for synthesis of pattern reconfigurable sparse arrays is presented. The method, based on the MMVs sparse recovery theory, is used for finding the common element positions and individual element excitations for multiple patterns in the guidance of collaborative synthesis strategy. A reasonable tradeoff between the number of active antenna elements and pattern characteristics can be achieved. Numerical examples show the high efficiency and robustness of the proposed method. Furthermore, the M-FOCUSS method is combined with the active element pattern technique so that the mutual coupling between real antennas can be taken into account in the synthesis process. The M-FOCUSS-Active algorithm is used to design an E-plane coupled microstrip antenna array. Measured results validate the effectiveness of this approach.

Synthesis of planar sparse arrays by perturbed compressive sampling framework

Recently, compressive sensing (CS) theory has been applied for synthesising maximally sparse arrays, in which the best subset of sampling element locations is chosen to compose a sparse array for matching a desired radiation pattern. However, their performances are strongly depended on the proper setting of the initial sampling locations, which are typically obtained by gridding the continuous array aperture. Such a setting is usually hard to handle for large planar array synthesis. To address this problem, a precision and effective method based on the perturbed compressive sampling (PCS) is proposed. Position perturbation variables are augmented to the traditional CS-based model, which allow continuous element placement. Then, a joint sparse recovery approach is used to optimise the excitations and position perturbations of the elements simultaneously. Moreover, the authors implement an extended PCS model with a secondary grid strategy to reduce the modelling error and the computational cost. The proposed design problem is solved with a general sparse recovery solver, named FOCal under-determined system solver. Numerical results show that the method yields a higher array sparsity, a faster computational speed and a better pattern matching accuracy than the existing CS-based methods.

Computational complexity reduction in nonuniform compressed sensing by multi-coset emulation

Single-channel nonuniform sampling (SNS) is a Compressed Sensing (CS) approach that allows sub-Nyquist sampling of frequency sparse signals. The relatively simple architecture, comprising one wide-band sampling channel, makes it an attractive solution for applications such as signal analyzers and telecommunications. However, a high computational cost of the SNS signal reconstruction is an obstacle for real-time applications. This paper proposes to emulate multi-coset sampling (MCS) in SNS acquisition as a means to decrease the computational costs. Such an emulation introduces performance-complexity tradeoffs due to the difference of the SNS and MCS models. We investigate these tradeoffs with numerical simulations and theoretical assessments of the reconstruction complexity in multi-band signal scenarios. These scenarios include different numbers, different widths and positions of the frequency bands and different levels of noise in the signals. For the SNS reconstruction, we consider the accelerated iterative hard thresholding algorithm; for the MCS reconstruction, the multiple signal classification and focal underdetermined system solver algorithms are used. The proposed emulation reduces the computational complexity up to several orders of magnitude. For one of the scenarios, the reconstruction quality slightly decreases. For the other scenarios, the reconstruction quality is either preserved or improved.

Rate of Convergence of the FOCUSS Algorithm.

Focal underdetermined system solver (FOCUSS) is a powerful method for basis selection and sparse representation, where it employs the [Formula: see text]-norm with p ∈ (0,2) to measure the sparsity of solutions. In this paper, we give a systematical analysis on the rate of convergence of the FOCUSS algorithm with respect to p ∈ (0,2) . We prove that the FOCUSS algorithm converges superlinearly for and linearly for usually, but may superlinearly in some very special scenarios. In addition, we verify its rates of convergence with respect to p by numerical experiments.